Existence of solutions to fourth-order differential equations with deviating arguments

摘要

In this paper, we consider fourth-order differential equations on a half-line with deviating arguments of the formu(4)(t)+q(t)f(t,[u(t)],[u′(t)],[u″(t)],u‴(t))=0$u^{(4)}(t)+q(t)f(t,[u(t)],[u'(t)],[u''(t)],u'''(t))=0$,0t+∞$0 t{+infty}$, with the boundary conditionsu(0)=A$u(0)=A$,u′(0)=B$u'(0)=B$,u″(t)−au‴(t)=θ(t)$u''(t)-au'''(t)=heta(t)$,−τ≤t≤0$-auleq{t}leq 0$;u‴(+∞)=C$u'''(+infty)=C$. We present sufficient conditions for the existence of a solution between a pair of lower and upper solutions by using Schäuder’s fixed point theorem. Also, we establish the existence of three solutions between two pairs of lower and upper solutions by using topological degree theory. An important feature of our existence criteria is that the obtained solutions may be unbounded. We illustrate the importance of our results through two simple examples.